HMMT 二月 2001 · 团队赛 · 第 11 题

英文解析

11. Define a ? = ( a − 1) / ( a + 1) for a 6 = − 1. Determine all real values N for which
    ( N ?)? = tan 15.
    Solution: Let x = N ?. Then ( x − 1) cos 15 = ( x + 1) sin 15. Squaring and rearranging
    √
    √
    2 3 22
    terms, and using the fact that cos 15 − sin 15 = cos 30 = , we have 3 x − 4 3 x + 3 = 0.
    √2
    √
    Solving, we find that x = 3 or . However, we may reject the second root because it 3
    √3
    √ √
    1+ x 1+ 3
    √
    yields a negative value for ( N ?)?. Therefore x = 3 and N = = = − 2 − 3 .
    1 − x
    1 − 3